To simplify the expression
(2+√x2−√x−4xx−4−2−√x2+√x):x−√x2√x−x
we can break it down step by step.
1. **Finding a Common Denominator**:
For the term
2+√x2−√x−2−√x2+√x
the common denominator is
(2−√x)(2+√x)=4−x. Therefore,
(2+√x)2−(2−√x)24−x
Simplifying the numerator:
(2+√x)2−(2−√x)2=(4+4√x+x)−(4−4√x+x)=8√x
Thus,
8√x4−x
2. **The Second Term**:
The second term is
−4xx−4=−4x−(4−x)=4x4−x
3. **Combining the First Two Terms**:
We combine:
8√x4−x+4x4−x=8√x+4x4−x
4. **Simplify the Expression**:
Now the overall expression becomes:
8√x+4x4−x−4xx−4
Converting the second part, we see:
4x4−x
Therefore our expression is:
8√x+4x−4x4−x=8√x4−x
5. **Divide by the Last Term**:
Now we need to divide by
x−√x2√x−x=−x−√xx−2√x=−1
6. **Final Expression**:
Thus, we have:
8√x4−x÷−1=−8√x4−x
In conclusion, the simplified form of the original expression is:
−8√x4−x